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Generalization of Scarpis's theorem on Hadamard matrices

Dragomir Z. Djokovic

Abstract

A $\{1,-1\}$-matrix $H$ of order $m$ is a Hadamard matrix if $HH^T=mI_m$, where $T$ is the transposition operator and $I_m$ the identity matrix of order $m$. J. Hadamard published his paper on Hadamard matrices in 1893. Five years later, Scarpis showed how one can use a Hadamard matrix of order $n=1+p$, $p\equiv 3 \pmod{4}$ a prime, to construct a bigger Hadamard matrix of order $pn$. In this note we show that Scarpis's construction can be extended to the more general case where $p$ is replaced by a prime power $q$.

Generalization of Scarpis's theorem on Hadamard matrices

Abstract

A -matrix of order is a Hadamard matrix if , where is the transposition operator and the identity matrix of order . J. Hadamard published his paper on Hadamard matrices in 1893. Five years later, Scarpis showed how one can use a Hadamard matrix of order , a prime, to construct a bigger Hadamard matrix of order . In this note we show that Scarpis's construction can be extended to the more general case where is replaced by a prime power .

Paper Structure

This paper contains 3 sections, 2 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Construction of $B$
  3. Proof that $B$ is a Hadamard matrix

Key Result

Theorem 1

Let $q\equiv 3 \pmod{4}$ be a prime power. If there exists a Hadamard matrix of order $n=1+q$ then there exists also a Hadamard matrix of order $qn$.

Theorems & Definitions (2)

  • Theorem 1
  • Corollary 1